Finite Group Databasebeta

← #14529689

Subgroups — #14529689

Order 2176.

296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587490C211yesmaximal, characteristic, complemented
2#11762818C211yesmaximal, characteristic, complemented
3#11762814C211yesmaximal, characteristic
4C2×#10587507C211yesmaximal, characteristic, complemented
5C2×#10587504C211yesmaximal, characteristic
6#11762764C211yesmaximal, characteristic, complemented
7C22×SD16×C17C211yesmaximal, characteristic
8C22×Q8×C17C2211yescharacteristic
9C22×D4×C17C2211yescharacteristic
10C2×SD16×C17—21no—
11C2×SD16×C17—21no—
12C2×SD16×C17—22no—
13C2×SD16×C17C414yescomplemented
14C22×C8×C17C2211yescharacteristic
15#10587612C2211yescharacteristic
16C2×#63223C2211yescharacteristic
17C2×#63223C2211yescharacteristic
18C2×#63223C2211yescharacteristic
19#10587507—22no—
20#10587506—22no—
21#10587504—22no—
22#10587503—22no—
23#10587490—22no—
24#10587486—21no—
25C8×Dic17—21no—
26C24×C17—21no—
27C2×Q8×C17—21no—
28C2×Q8×C17—21no—
29C2×Q8×C17—21no—
30C2×Q8×C17—21no—
31C2×Q8×C17—21no—
32C2×Q8×C17C2×C412yes—
33C2×D4×C17—21no—
34C2×D4×C17—21no—
35C2×D4×C17—21no—
36C2×D4×C17—21no—
37C2×D4×C17—21no—
38C2×D4×C17C2×C412yes—
39C22×C4×C17—21no—
40C22×C4×C17C2311yescharacteristic
41SD16×C17—24no—
42SD16×C17—24no—
43C2×C8×C17C2×C412yes—
44C2×C8×C17D411yescharacteristic
45C2×C8×C17D411yescharacteristic
46C2×C8×C17D412yes—
47C22⋊Dic17—21no—
48C4⋊Dic17D412yes—
49C4×Dic17—21no—
50#63223—21no—
51#63223—21no—
52#63223—21no—
53#63223—21no—
54#63223—21no—
55#63223D412yes—
56C2×#3622—21no—
57C2×#3622—21no—
58C2×#3622—21no—
59C23×C17—22no—
60C23×C17—41no—
61C23×C17—41no—
62C23×C17—82no—
63C23×C17#4111yescharacteristic
64Q8×C17—22no—
65Q8×C17—22no—
66Q8×C17—41no—
67Q8×C17—41no—
68D4×C17—22no—
69D4×C17—22no—
70D4×C17—41no—
71D4×C17—41no—
72C2×C4×C17—22no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17C22×C411yescharacteristic
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17#4111yescharacteristic
81C2×Dic17—22no—
82C8×C17C22⋊C412yes—
83C8×C17C22⋊C412yes—
84#3622—22no—
85#3622—41no—
86#3622—41no—
87#1977—171nomaximal, Sylow
88C22×C17—22no—
89C22×C17—42no—
90C22×C17—82no—
91C22×C17—82no—
92C22×C17—82no—
93C22×C17C4×D411yescharacteristic
94C22×C17#12111yescharacteristic
95C22×C17#12411yescharacteristic
96C4×C17—42no—
97C4×C17C2×C22⋊C411yescharacteristic
98C4×C17C4×D411yescharacteristic
99C4×C17#12111yescharacteristic
100C4×C17#12711yescharacteristic
101Dic17—42no—
102C22×SD16D1711yescharacteristic
103C2×#108—171no—
104#419—171no—
105#418—171no—
106C2×#101—171no—
107C2×#100—171no—
108#405—171no—
109C2×C17—21no—
110C2×C17—21no—
111C2×C17—82no—
112C2×C17#39011yescharacteristic
113C2×C17#47112yes—
114C22×Q8C2×D1711yescharacteristic
115C22×D4C2×D1711yescharacteristic
116C2×SD16—21no—
117C2×SD16—21no—
118C2×SD16—22no—
119C2×SD16Dic1714yescomplemented
120C2×M16—171no—
121C2×M16—171no—
122C2×M16—171no—
123C22×C8C2×D1711yescharacteristic
124#117—171no—
125#108—342no—
126#103—342no—
127#102—342no—
128#101—342no—
129#100—342no—
130#97—341no—
131C4×C8—341no—
132C17#197711yescharacteristic, complemented, Sylow
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×Dic1712yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Dic1712yes—
146C22×C4—21no—
147C22×C4C22×D1711yescharacteristic
148SD16—24no—
149SD16—24no—
150M16—172no—
151M16—341no—
152M16—341no—
153M16—341no—
154M16—341no—
155M16—341no—
156C2×C8—341no—
157C2×C8—341no—
158C2×C8—341no—
159C2×C8C2×Dic1712yes—
160C2×C8C22⋊D1711yescharacteristic
161C2×C8C22⋊D1711yescharacteristic
162C2×C8C22⋊D1712yes—
163C4⋊C4—172no—
164C22⋊C4—341no—
165C42—341no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#6325411yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—342no—
183C2×C4D4×D1711yescharacteristic
184C2×C4D4×D1711yescharacteristic
185C2×C4#6325411yescharacteristic
186C2×C4C22×Dic1711yescharacteristic
187C2×C4C2×C22⋊D1711yescharacteristic
188C2×C4C2×C22⋊D1711yescharacteristic
189C8—342no—
190C8—681no—
191C8—681no—
192C8C22⋊Dic1712yes—
193C8C22⋊Dic1712yes—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22D4×Dic1711yescharacteristic
200C22#1058762211yescharacteristic
201C22#1058762411yescharacteristic
202C4—42no—
203C4—682no—
204C4D4×Dic1711yescharacteristic
205C4#1058762411yescharacteristic
206C4#1058762611yescharacteristic
207C4C2×C22⋊Dic1711yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#1176271211yescharacteristic
212C2#1176306912yes—
subgroupG/H
centerC22#10587624
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC22×SD16×C17C2
socleC22×C17#121
prime radicalC22×D4×C17C22
cosocleC22×C4×C17C23
hypercenterC22×SD16D17
hypocenterC17#1977
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#10587490C2semidirect
2. #11762818C2semidirect
4. C2×#10587507C2semidirect
6. #11762764C2semidirect
13. C2×SD16×C17C4semidirect
119. C2×SD16Dic17semidirect
132. C17#1977semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#105874902
#117628182
#117628142
C2×#105875072
C2×#105875042
#117627642
C22×SD16×C172
#197717